When riding in a minecart on 1.5.1 the player is seated higher than he should be.
This can sometimes be worked around by getting into a minecart on a hill, but getting into a minecart on a flat surface almost always produces this result.
I'm attaching a picture of the correct behavior, and the incorrect behavior when getting into the same minecart on the same rail.
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Is it just visual from 3rd person (appears fine to rider), 3rd and first (glitched to both viewer and rider), or both as well as physical (can't fit into 1x2 tunnel either)?

It affects 1st person as well. You can travel through a 2x1 tunnel just fine, but your head is right up against the top block.
This made my nether rail (rail with glass directly above it) vulnerable as a Ghast had line of sight through the top of the glass.
Some times you will sink down a little in the cart, but not to the previous level. You have to get into the cart on a slope to get to this level.
I can send more screen caps with debug info pulled up if you'd like.

Screenshots showing the debug coordinates in 1.5 and 1.5.1, while standing on the rail, and sitting in the cart.
1.5 Standing on Rail: y=114.000
1.5 Sitting in Cart: y=113.700
1.5.1 Standing on Rail: y=114.000
1.5.1 Sitting in Cart: y=114.044

I consistently get this issue when entering a minecart too. My character sits down but is too high so when I try to click the 'start' button (even one that's at head height, a block above the track) I actually get out of the minecart. However if I wait ~2 seconds then my character sinks down into the cart and allows me to click 'start' just as it worked in 1.4.7 and previously.

I've noticed the same behavior as Mark, but I think it is subtly different from what Sam reported.
I've attached another screenshot from 1.5.1 showing the debug coordinates when entering the cart:
1.5.1 Entering cart: y=114.394
This doesn't always happen when entering a cart, and after a few seconds the player lowers to y=114.044.
I've also noticed that if you place a cart and relog, when you enter that cart the player position is always correct: 113.700. I'm guessing there's a cart collision box update bug that's causing this.
Edit: Accidentally referred to the OP as Stephen, not Sam.

Confirmed.

@John I didn't report anything, I was just asking for clarification.

I have this issue too. It makes our minecart system very tedious to use. I seem to remember this bug existing a while back (beta 1.7.3?) but can't remember when it was resolved.

I would guess that the Minecart collision Box is placed too high, as when you enter, everything is moved up those 0.344 blocks. So the lowest spot you can click on is 0.344 blocks higher than in 1.5. The problem why this doesn't make sense is that if you relog (with the cart placed before or being in the cart) after 1-3 seconds everything is normal again.
Oh yes, minecraft is weird. 😉

They should REALLY fix this because it is so annoying. Please Mojang! Please!

after a few seconds you sink back down a little then after riding for some time you sink down to the correct level

if you want it fixed vote for it

Can no longer observe this issue in 13w21b.
Edit: Still seen this issue in 13w21b

I can get it to happen for an instant.

I'll try it tomorrow.

Thanks Jeb! You're awesome <3
God, I'm so glad that this is fixed.